|
Arc Topology
In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valuation rings. This topology was introduced by and studied further by , who introduced the name ''v''-topology, where ''v'' stands for valuation. Definition A universally subtrusive map is a map ''f'': ''X'' → ''Y'' of quasi-compact, quasi-separated schemes such that for any map ''v'': Spec (''V'') → ''Y'', where ''V'' is a valuation ring, there is an extension (of valuation rings) V \subset W and a map Spec ''W'' → ''X'' lifting ''v''. Examples Examples of ''v''-covers include faithfully flat maps, proper surjective maps. In particular, any Zariski covering is a ''v''-covering. Moreover, universal homeomorphisms, such as X_ \to X, the normalisation of the cusp, and the Frobenius in positive characteristic are ''v''-coverings. In fact, the perfection X_ \to X of a sche ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation Ring
In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' such that either ''x'' or ''x''−1 belongs to ''D'' for every nonzero ''x'' in ''F'', then ''D'' is said to be a valuation ring for the field ''F'' or a place of ''F''. Since ''F'' in this case is indeed the field of fractions of ''D'', a valuation ring for a field is a valuation ring. Another way to characterize the valuation rings of a field ''F'' is that valuation rings ''D'' of ''F'' have ''F'' as their field of fractions, and their ideals are totally ordered by inclusion; or equivalently their principal ideals are totally ordered by inclusion. In particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially ordered by dominance or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H Topology
In algebraic geometry, the ''h'' topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"topologies, such as the ''qfh'' and ''cdh'' topologies. It has subsequently been used by Beilinson to study p-adic Hodge theory, in Bhatt and Scholze's work on projectivity of the affine Grassmanian, Huber and Jörder's study of differential forms, etc. Definition Voevodsky defined the ''h'' topology to be the topology associated to finite families \ of morphisms of finite type such that \amalg U_i \to X is a universal topological epimorphism (i.e., a set of points in the target is an open subset if and only if its preimage is open, and any base change also has this property). Voevodsky worked with this topology exclusively on categories Sch^_ of schemes of finite type over a ''Noetherian'' base scheme S. Bhatt-Scholze define the ''h'' topology on the category Sch^_ of schemes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amitsur Complex
In algebra, the Amitsur complex is a natural complex associated to a ring homomorphism. It was introduced by . When the homomorphism is faithfully flat, the Amitsur complex is exact (thus determining a resolution), which is the basis of the theory of faithfully flat descent. The notion should be thought of as a mechanism to go beyond the conventional localization of rings and modules. Definition Let \theta\colon R \to S be a homomorphism of (not-necessary-commutative) rings. First define the cosimplicial set C^\bullet = S^ (where \otimes refers to \otimes_R, not \otimes_) as follows. Define the face maps d^i\colon S^ \to S^ by inserting 1 at the ''i''-th spot: :d^i(x_0 \otimes \cdots \otimes x_n) = x_0 \otimes \cdots \otimes x_ \otimes 1 \otimes x_i \otimes \cdots \otimes x_n. Define the degeneracies s^i\colon S^ \to S^ by multiplying out the ''i''-th and (''i'' + 1)-th spots: :s^i(x_0 \otimes \cdots \otimes x_n) = x_0 \otimes \cdots \otimes x_i x_ \otimes \cdots \o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Field
In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is separable. * Every algebraic extension of ''k'' is separable. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , every element of ''k'' is a ''p''th power. * Either ''k'' has characteristic 0, or, when ''k'' has characteristic , the Frobenius endomorphism is an automorphism of ''k''. * The separable closure of ''k'' is algebraically closed. * Every reduced commutative ''k''-algebra ''A'' is a separable algebra; i.e., A \otimes_k F is reduced for every field extension ''F''/''k''. (see below) Otherwise, ''k'' is called imperfect. In particular, all fields of characteristic zero and all finite fields are perfect. Perfect fields are significant because Galois theory over these fields becomes simpler, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Complex
Exact may refer to: * Exaction, a concept in real property law * '' Ex'Act'', 2016 studio album by Exo * Schooner Exact, the ship which carried the founders of Seattle Companies * Exact (company), a Dutch software company * Exact Change, an American independent book publishing company * Exact Editions, a content management platform Mathematics * Exact differentials, in multivariate calculus * Exact algorithms, in computer science and operations research * Exact colorings, in graph theory * Exact couples, a general source of spectral sequences * Exact sequences, in homological algebra * Exact functor, a function which preserves exact sequences See also * *Exactor (other) *XACT (other) XACT can refer to: * Cross-platform Audio Creation Tool, an audio system developed by Microsoft * IP-XACT, an XML-based standard covering electronic components * Xact Radio, a former brand used by broadcaster WNTY in the United States See also ... * EXACTO, a sniper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Topologies On The Category Of Schemes
The most fundamental item of study in modern algebraic geometry is the category of schemes. This category admits many different Grothendieck topologies, each of which is well-suited for a different purpose. This is a list of some of the topologies on the category of schemes. * ''cdh'' topology A variation of the h topology * Étale topology Uses etale morphisms. * fppf topology Faithfully flat of finite presentation * fpqc topology Faithfully flat quasicompact * ''h'' topology Coverings are universal topological epimorphisms * v-topology (also called universally subtrusive topology): coverings are maps which admit liftings for extensions of valuation rings * ' topology A variation of the Nisnevich topology * Nisnevich topology Uses etale morphisms, but has an extra condition about isomorphisms between residue fields. * ''qfh'' topology Similar to the h topology with a quasifiniteness condition. * Zariski topology Essentially equivalent to the "ordinary" Zariski topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
